Convention Paper

Audio Engineering Society

Convention Paper Presented at the 123rd Convention 2007 October 5–8 New York, NY The papers at this Convention have been selected on the basis of a submitted abstract and extended precis that have been peer reviewed by at least two qualified anonymous reviewers. This convention paper has been reproduced from the author’s advance manuscript, without editing, corrections, or consideration by the Review Board. The AES takes no responsibility for the contents. Additional papers may be obtained by sending request and remittance to Audio Engineering Society, 60 East 42nd Street, New York, New York 10165-2520, USA; also see www.aes.org. All rights reserved. Reproduction of this paper, or any portion thereof, is not permitted without direct permission from the Journal of the Audio Engineering Society.

Modelling of Nonlinearities in Electrodynamic Loudspeakers Delphine Bard1 , and G¨ oran Sandberg1 1

Eng. Acoustics, University of Lund, 221 00 Lund, Sweden

Correspondence should be addressed to Delphine Bard ([email protected]) ABSTRACT This paper proposes a model of the non-linearities in an electro-dynamic loudspeaker based on Volterra series decomposition and taking into account the thermal effects affecting the electrical parameters when temperature increases. This model will be used to predict nonlinearities taking place in a loudspeaker and their evolution as the loudspeaker is used for a long time and/or at high power rates and its temperature increases. A temperature increase of the voice coil will cause its series resistance value to increase, therefore reducing the current flowing in the loudspeaker. This phenomenon is known as power compression.

1. INTRODUCTION Extensive research has been carried out to develop loudspeaker characterization methods. A nonexhaustive set of them can be found in [10], [11], [12]. However, the characterization of loudspeakers and the compensation of their nonlinearities always requires new methods, in order to cover a wider range of the nonlinearities that we can encounter. The method presented here, based on Volterra kernels and taking into account the thermal effects affecting the electrical parameters when temperature increases, allows this deeper investigation, since it covers the distortions resulting from the

mutual effect of several components on each other. The method involves an excitation constituted by a sum of sinus at different frequencies. For the weakly nonlinear loudspeakers, and over their whole frequency domain, it provides with detailed information about nonlinearities. The characterization method takes the electroacoustic system into consideration as a black box whose properties are only determined by the transformations occurring between the input and the output. The measured kernels would then be those characterizing this box in its whole, as suggested by fig-

Delphine Bard AND G¨ oran Sandberg

Nonlinearities Modelling

Fig. 1: Electroacoustic device simplification as a “black box”. ure 1. In the present work, the method is described from the theoretical point of view. 2. THERMAL EFFECTS The voice coil of a loudspeaker is characterized, among other parameters, by its series resistance RE . The current flowing through this resistive load results in some heat dissipation. The dissipated power is equal to the product of the current flowing through the coil by the voltage across it. In our case, it is easier to consider it as the product of the series resistance by the squared current. The power dissipation causes the coil temperature to increase accordingly. The coil is surrounded by other loudspeaker elements like the magnet, the frame, the suspension, the cone. Those elements are all mechanically and thermally coupled. Therefore, any temperature increase of the coil will affect the temperature of the whole loudspeaker structure, although this effect will take some time to be noticed. Furthermore, the loudspeaker is in contact with the ambient air, and heat exchanges occur here as well. As it has been shown by Behler et. al. [14], this complex thermal system can be modelled quite easily by considering that the different elements (voice coil, magnet, speaker structure) are characterized by their thermal capacitance and resistance. Finally, the interface between the speaker surface and the surrounding air must be taken into account as well by considering the ambient temperature. Usually, the system is modelled by two to three thermal elements. Each of them is characterized by its thermal resistance and capacitance. We will use one element for the coil and a second one for the magnet. A third element is sometimes used to represent the cabinet in which the loudspeaker is enclosed. We do not use this latter for the present study, as the very slow temperature changes it causes are not very relevant for this study. However, the results we discuss

Fig. 2: Thermal model. would be still valid with any thermal model. The figure 2 represents the model we use. The input parameter of the thermal model is the heat power Q(t) dissipated in the voice coil. The first element is characterized by a relatively small thermal capacitance CCoil , due to the low mass of the coil itself. This means that it will heat up very quickly as some current flows through it, and cool down quickly as well when current stops to flow. The thermal resistance RCoil represents how easily the heat can be transferred from the coil to the next element, that is the magnet. The coil is a mobile element that must be surrounded by the magnetic field produced by the magnet. Therefore, there is no contact between the magnet and the coil, but the magnetic gap has to remain as small as possible in order to have the maximum efficiency of the magnetic coupling. This tiny gap is responsible for a relatively large thermal resistance value, causing the magnet to progressively heat up when the coil temperature increases. The magnet is a massive piece of magnetic material, mechanically and thermally coupled with the bulk of the loudspeaker. Therefore, it has a large thermal capacitance value. The consequence is that it will heat up slowly, but will also take a long time to cool down after the current stops flowing through the coil. In addition, the heat transfer between the loudspeaker surface and the ambient air is rather inefficient. Manufacturers of high power rated loudspeakers try to thermal resistance of their transducers by increasing the exchange surface with heat sinks. However, the interface between the surface and the air remains a major bottleneck in the heat flow. The figure 3 represents the evolution of the voice-coil and magnet temperature as a function of the time

AES 123rd Convention, New York, NY, 2007 October 5–8 Page 2 of 7


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Fig. 3: Evolution of the temperature of the voice coil and magnet in a loudspeaker fed with a constant power. when a constant power is supplied to a loudspeaker. The parameter values have been chosen from the literature, since the parameters of the loudspeaker used for this study are not known. They are summarized in the table 1. The loudspeaker behavior has been calculated with a constant power of 25W, 50W and 100W. The figure shows the evolution of the voice coil and magnet temperatures. We can notice the very large time constant of the magnet, due to its high thermal capacitance. It is interesting to notice as well how fast the voice coil temperature can rise. Even at a moderate input power of 25W, the coil temperature can reach values of 750 C after approximately one hour of continuous excitation. At higher power rates, the temperature rise can be very fast. After one hour at 50W, the coil reaches 125oC. This temperature is however reached in less than 20s at a constant power rate of 100W. After only 40s at the same rate, the temperature reaches 180oC, and this latter can reach close to 250o C after several hours. At the same time, the magnet temperature will reach temperatures above 50o C. We will see later the effects that those temperature changes can have on the electrical parameters. 2.1. Electrical model We use the classical lumped element loudspeaker

model represented on the figure 4. The voice coil series resistance is represented by RE whereas its inductive counterpart is represented by LE . The elements REM , CEM and LEM are used to take into account the mechanical properties of the transducer. The electrical parameters of the woofer used for this study are summarized in the table 1. 2.2. Consequences of the voice coil temperature changes The wire used for the voice coil is generally made of Copper or Aluminium. Its resistivity varies slightly with the temperature. However, when we are concerned with a temperature increase of 100o C or more, the corresponding variation of the series resistance can not be neglected any more. When the loudspeaker is driven in voltage mode, the consequence of the series resistance increase is a decrease of the power available to the loudspeaker. This phenomenon is known as power compression. The other way round, when the loudspeaker is driven in current mode, the increase of the series resistance causes the voltage across the loudspeaker unit to increase, and so does the loudspeaker power. We use a quadratic function to model the resistivity change as a function of the temperature.

R(θ0 + ∆θ) = R0 · 1 + α0 · ∆θ + β0 · ∆θ2

(1)

where R0 is the resistance value at the reference temperature θ0 (usually the ambient temperature, 25o ), and ∆θ is the difference between the actual coil temperature θ and the reference temperature. For a Copper wire, α0 = 3.93 · 10−3 K −1 and β0 = 0.7 · 10−6 K −2 .



Parameter Name Total moving mass Mechanical compliance Mechanical resistance Free air resonance Mechanical quality factor Force factor Voice coil inductance Voice coil series resistance Voice Coil thermal resistance Voice Coil thermal capacitance Magnet thermal resistance Magnet thermal capacitance


Symbol MMS CM RM FSA Qms Bl Le Re RCoil CCoil RMagnet CMagnet

Value 0.037 kg 1.013 · 10−3 m · N 1.6 N · s · m−1 26 Hz 3.8 12 T · m 2.6 mH 5.8 Ω 1.69 K · W −1 10.8 W · s · K −1 0.429 K · W −1 6360 W · s · K −1

Table 1: Parameters of the woofer referenced SEAS 25F-EWX

For the electrical part of the modelling, we solve the following system of coupled differential equations:  ULS      UEM ICEM    I   REM IE

= = = = =

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The electrical voltages and currents are evaluated for each time step by using an explicit solving method. The power PE fed to the loudspeaker voice coil can be deduced from RE and IE . This power value is used to deduce the temperature of the coil and of the magnet. The voice coil temperature is then used to derive RE (θ). 3. VOLTERRA SERIES Volterra series permit the characterization of the non-linearities laws governing a weakly non-linear system. A Volterra kernel of nth order is a descriptor of the temporal behavior of the device at the nth order. It is represented by a continuous function of n variables which correspond to n simultaneously interacting input frequencies, and noted hn . Each kernel of the device contributes to the response r(t) of the device to an excitation e(t). The global response of the device is constituted by the sum of the contributions due to each kernel, as illustrated on the figure (5). The τi represent here the temporal integration variables.

Non-linear

Fig. 5: System decomposed by Volterra series. In the frequency domain, the first order kernel is constituted by the linear transfer function, while the next kernels constitute its generalization at higher orders. An efficient determination of a Volterra kernel requires an excitation with the appropriate component number, frequencies, magnitudes and phases. The Volterra kernels are deduced from the measured response. The choice of the frequencies is based on the works of Boyd et al.[1] and Chua et al.[2]. A nth order kernel is a continuous function of n variables, which can be approximated by a discrete set of measurement points, at the condition that their density is large enough. Successive measurements at increasing excitation levels are required to determine each of these points. We can however obtain




quickly a large amount of points for each measure by increasing the number of components in an excitation signal. In the case where several combinations of component frequencies lead to the interactions at the same frequency, we talk about frequency confusion. In order to guarantee the effectiveness and accuracy of the method, we will take care to avoid this phenomenon by choosing harmonic components of a fundamental in such a manner that the whole set of their possible combination does only contain unique terms. As a matter of example, For the determination of the second order kernel, a sufficient condition to avoid confusion is to group the frequencies in two sets of harmonic frequencies, with fundamentals (P, Q) which frequencies are prime. For a third order kernel, the choice of three harmonic excitation groups, each containing m harmonics, respectively multiples of (P, Q, R) = (Rmin − 1, Rmin , Rmin + D − 1) with Rmin = (2m + 1)2 and D = 2(m + 1), ensures a repartition with no confusion. 3.1. System resolution A system which order is assumed to be n requires the previously defined excitation to be applied at n excitation levels αi . The system response ri (t) to each of these excitations is composed by the sum of the components at the orders 1 to n. These components can be identified from the yi (t) measured signals by resolving the matrix system 1: R = A.Y

(2)

where A is the Van Der Monde Matrix built with the αi coefficients, and R and Y respectively contain the ri (t) and yi (t). The αi coefficients are chosen in such a manner that we have the best available accuracy, while staying in the required nonlinearity domain for the device under test. For the orders 1 to 4, the optimum excitation levels have been computed from [2]: (1), (1; -1), (1; 0.5; -1), (1, 0.634, -0.634, -1). 3.2. Kernel extraction in the frequency domain A Fourrier transform of the yn (t) and of the excitation signals en (t) allows us to determine the spectra Yn (jω) and En (jω), that will be used in the following equation to extract the kernels. The functions Hn (jω) represent the Volterra kernels in the frequency domain.

Hn (jωi1 , . . . , jωin ) =

Pn Yn (j k=1 ωik ) E(jωi1 ), . . . , E(jωin )

(3)

4. RESULTS A woofer 25F-EWX from the brand SEAS has been chosen as a model for the calculation. The loudspeaker Volterra kernels are calculated by solving the differential equation system. Different temperatures were chosen, ranging from 25o C (ambient temperature) up to 250o C. The left side of the figure 6 represents the magnitude of the first order kernel (transfer function) at different temperature. We can observe a dramatic decrease of the bandwidth when the temperature is increasing. Furthermore, the maximum magnitude is reduced as well for higher temperatures. The right side of the figure represents the phase of the same kernel under the same temperature conditions. Here again, we can notice a quite large difference in the phase response of the loudspeaker over the temperature range. The figure 7 represents the second-order Volterra kernels at different temperatures. We can notice a change in the shape of the surface map when the temperature is increased. The surface looks more smooth for higher temperature and this effect can be noticed especially at higher frequencies. From the decomposition in Volterra kernels, we can assume that the distortion products will also depend upon the temperature of the voice coil. Therefore, it is strongly suggested to take into account the heating of the voice coil in methods of characterization and compensation of loudspeaker nonlinearities. 5. CONCLUSION An investigation of the loudspeaker response in terms of Volterra kernels has been done. The focus has been put on the temperature dependence. This work is an extension of previous investigation about loudspeaker characterization using Volterra kernels ([6][7][8][9]). The model used here is still basic, since we use only the classical lumped-element circuit as a basis for the calculations. However, it has been proved that this approach is valid to understand the




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Fig. 6: First order Volterra kernel of the loudspeaker at different temperatures. The magnitude is represented on the left side and the phase on the right. nonlinear effects of a temperature increase on the loudspeaker response. Further extension of this work is planned by using more complex models. Then, a correlation with measured data will probably be relevant. 6. REFERENCES [1] S.Boyd,Y S.Tang, L.O.Chua “Measuring Volterra Kernels” IEEE Transactions on circuits and systems, vol. Cas-30, no8, August 1983 [2] L.O.Chua, Y. Liao “Measuring Volterra Kernels II” International journal of circuit theory and applications,vol.17,151-190 (1989) [3] M.Rossi “Traite d’electricite: Electroacoustique” Presses Polytechniques romandes, Lausanne, 1986 [4] M.Schetzen “The Volterra and Wiener theories of nonlinear systems” Wiley, New York, 1980 [5] L. Beranek “Acoustic” electrical and electronic engineering series, Graw-Hill, 1954 [6] D.Bard “Nonlinearities characterizations” AES 117th Convention, San Franscico, USA, October 2004.

[7] D.Bard “Characterization of nonlinearities of electroacoustic devices using Volterra kernels” ICSV12, Lisbon, Portugal, July 2005. [8] D.Bard “Compensation des non-linéarites des systèmes haut-parleur a` pavillon” Ph.D. thesis ,number 3317, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, July 2005. [9] D.Bard, M. Rossi and M.D. Nobile “Compensation of nonlinearities of horn loudspeakers”, presented at the 119th convention, New-York, USA, October 2005 [10] Angelo Farina, Emanuele Ugolotti, Alberto Bellini, Gianfranco Cibelli, Carlo morandi, “Inverse numerical filters for linearisation of loudspeaker’s response”. [11] Wolfgang Klippel “Compensation for nonlinear distorsion of horn loudspeakers by Digital Signal Processing”, J.A.E.S., November 1996, Vol 44, pp. 964-972. [12] A.J.M. Kaizer “Modeling of the nonlinear response of an electrodynamic loudspeaker by a Volterra series expansion”, J.A.E.S., June 1987, Vol. 35, pp. 421-433.



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Fig. 7: Second order Volterra kernel of the loudspeaker at three different temperatures - Magnitude. On the top: 25o C on the left and 125oC on the right. At the bottom: 250oC [13] Alex Voishvillo, “Assesment of Nonlinearity in Transducers and Sound Systems - From THD to Perceptual Models” presented at the 121st convention, San-Francisco, CA, USA, 2006 October 5–8. [14] Gottfried K. Behler, Armin Bernhard, “Measuring method to derive the lumped elements of the loudspeaker thermal equivalent circuit” presented at the 104th convention, Amsterdam, Netherland, 1998 May 16-19.
